To find the sum of \( 6.3 \times 10^3 \) and \( 2.41 \times 10^1 \), we first need to express both numbers in scientific notation with the same exponent.
The number \( 2.41 \times 10^1 \) can be converted to the same exponent as \( 6.3 \times 10^3 \):
\[ 2.41 \times 10^1 = 2.41 \times 10^1 \times \frac{10^2}{10^2} = 2.41 \times 10^3 \times 0.01 = 0.241 \times 10^3 \]
Now we can add \( 6.3 \times 10^3 \) and \( 0.241 \times 10^3 \):
\[ 6.3 \times 10^3 + 0.241 \times 10^3 = (6.3 + 0.241) \times 10^3 = 6.541 \times 10^3 \]
Thus, the sum of \( 6.3 \times 10^3 \) and \( 2.41 \times 10^1 \) is
\[ \boxed{6.541 \times 10^3} \]